Problem: $\dfrac{ 9a + b }{ 6 } = \dfrac{ -6a + 7c }{ -5 }$ Solve for $a$.
Answer: Multiply both sides by the left denominator. $\dfrac{ 9a + b }{ {6} } = \dfrac{ -6a + 7c }{ -5 }$ ${6} \cdot \dfrac{ 9a + b }{ {6} } = {6} \cdot \dfrac{ -6a + 7c }{ -5 }$ $9a + b = {6} \cdot \dfrac { -6a + 7c }{ -5 }$ Multiply both sides by the right denominator. $9a + b = 6 \cdot \dfrac{ -6a + 7c }{ -{5} }$ $-{5} \cdot \left( 9a + b \right) = -{5} \cdot 6 \cdot \dfrac{ -6a + 7c }{ -{5} }$ $-{5} \cdot \left( 9a + b \right) = 6 \cdot \left( -6a + 7c \right)$ Distribute both sides $-{5} \cdot \left( 9a + b \right) = {6} \cdot \left( -6a + 7c \right)$ $-{45}a - {5}b = -{36}a + {42}c$ Combine $a$ terms on the left. $-{45a} - 5b = -{36a} + 42c$ $-{9a} - 5b = 42c$ Move the $b$ term to the right. $-9a - {5b} = 42c$ $-9a = 42c + {5b}$ Isolate $a$ by dividing both sides by its coefficient. $-{9}a = 42c + 5b$ $a = \dfrac{ 42c + 5b }{ -{9} }$ Swap signs so the denominator isn't negative. $a = \dfrac{ -{42}c - {5}b }{ {9} }$